Steve Awodey
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.001.0001
- Subject:
- Mathematics, Algebra
This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible ...
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This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.Less
This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.
D. Huybrechts
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199296866
- eISBN:
- 9780191711329
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296866.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this ...
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This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.Less
This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.
John L. Bell
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568520
- eISBN:
- 9780191717581
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568520.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This is the third edition of a well-known graduate textbook on Boolean-valued models of set theory. The aim of the first and second editions was to provide a systematic and adequately motivated ...
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This is the third edition of a well-known graduate textbook on Boolean-valued models of set theory. The aim of the first and second editions was to provide a systematic and adequately motivated exposition of the theory of Boolean-valued models as developed by Scott and Solovay in the 1960s, deriving along the way the central set theoretic independence proofs of Cohen and others in the particularly elegant form that the Boolean-valued approach enables them to assume. In this edition, the background material has been augmented to include an introduction to Heyting algebras. It includes chapters on Boolean-valued analysis and Heyting-algebra-valued models of intuitionistic set theory.Less
This is the third edition of a well-known graduate textbook on Boolean-valued models of set theory. The aim of the first and second editions was to provide a systematic and adequately motivated exposition of the theory of Boolean-valued models as developed by Scott and Solovay in the 1960s, deriving along the way the central set theoretic independence proofs of Cohen and others in the particularly elegant form that the Boolean-valued approach enables them to assume. In this edition, the background material has been augmented to include an introduction to Heyting algebras. It includes chapters on Boolean-valued analysis and Heyting-algebra-valued models of intuitionistic set theory.
Sam Glucksberg
- Published in print:
- 2001
- Published Online:
- January 2008
- ISBN:
- 9780195111095
- eISBN:
- 9780199872107
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195111095.001.0001
- Subject:
- Psychology, Cognitive Psychology
The book presents a comprehensive account of how people understand metaphors and idioms in everyday discourse. Traditionally, figurative language has been considered to be derived from and more ...
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The book presents a comprehensive account of how people understand metaphors and idioms in everyday discourse. Traditionally, figurative language has been considered to be derived from and more complex than literal language. The book presents an alternative view, arguing that figurative language makes use of the same kinds of linguistic and pragmatic operations that are used for literal language. A new theory of metaphor comprehension integrates linguistic, philosophical, and psychological perspectives to account for figurative language use. The theory's central tenet is that everyday conversational metaphors are used spontaneously to create new concepts and categories. Metaphor is special only in the sense that metaphorical categories are salient examples of the things that they represent. These categories get their names from the best examples of those categories. Thus, the literal “shark” can be a metaphor for any vicious and predatory creature. Because the same term, “shark”, is used for both its literal referent and for the metaphorical category, as in “my lawyer is a shark”, such terms have dual-reference. In this way, metaphors simultaneously refer to the abstract metaphorical category and to the most salient literal exemplar of that category, as in the expression “boys (literal) will be boys (metaphorical)”. The book concludes with a comprehensive treatment of idiom use, and an analysis and critique (written by Matthew McGlone) of conceptual metaphor in the context of how people understand both conventional and novel figurative expressions.Less
The book presents a comprehensive account of how people understand metaphors and idioms in everyday discourse. Traditionally, figurative language has been considered to be derived from and more complex than literal language. The book presents an alternative view, arguing that figurative language makes use of the same kinds of linguistic and pragmatic operations that are used for literal language. A new theory of metaphor comprehension integrates linguistic, philosophical, and psychological perspectives to account for figurative language use. The theory's central tenet is that everyday conversational metaphors are used spontaneously to create new concepts and categories. Metaphor is special only in the sense that metaphorical categories are salient examples of the things that they represent. These categories get their names from the best examples of those categories. Thus, the literal “shark” can be a metaphor for any vicious and predatory creature. Because the same term, “shark”, is used for both its literal referent and for the metaphorical category, as in “my lawyer is a shark”, such terms have dual-reference. In this way, metaphors simultaneously refer to the abstract metaphorical category and to the most salient literal exemplar of that category, as in the expression “boys (literal) will be boys (metaphorical)”. The book concludes with a comprehensive treatment of idiom use, and an analysis and critique (written by Matthew McGlone) of conceptual metaphor in the context of how people understand both conventional and novel figurative expressions.
Claude Hagège
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199575008
- eISBN:
- 9780191722578
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575008.003.0006
- Subject:
- Linguistics, Syntax and Morphology
This concluding chapter begins with a review of discussions in the preceding chapters. It then discusses adpositions as morpholexical units shedding light on a theory of linguistic categories; ...
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This concluding chapter begins with a review of discussions in the preceding chapters. It then discusses adpositions as morpholexical units shedding light on a theory of linguistic categories; adpositions as midpoints and images of language leaks and diachronic drifts; and morphology as the most linguistic component of human languages.Less
This concluding chapter begins with a review of discussions in the preceding chapters. It then discusses adpositions as morpholexical units shedding light on a theory of linguistic categories; adpositions as midpoints and images of language leaks and diachronic drifts; and morphology as the most linguistic component of human languages.
Paul Crowther
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199579976
- eISBN:
- 9780191722615
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199579976.001.0001
- Subject:
- Philosophy, Aesthetics, History of Philosophy
This book explains the perceptual knowledge involved in aesthetic judgements. It does so by linking Kant's aesthetics to a critically upgraded account of his theory of knowledge. This upgraded theory ...
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This book explains the perceptual knowledge involved in aesthetic judgements. It does so by linking Kant's aesthetics to a critically upgraded account of his theory of knowledge. This upgraded theory emphasizes conceptual and imaginative structures, which Kant terms respectively, ‘categories’ and ‘schemata’. By describing examples of aesthetic judgement, it is shown that these judgements must involve categories and fundamental schemata (even though Kant himself, and most other commentators, have not fully appreciated the fact). It is argued, in turn, that this shows the aesthetic to be not just one kind of pleasurable experience amongst others, but one based on factors necessary to objective knowledge and personal identity, and one, indeed, which plays a role in how these capacities develop. The explanation of how individual aesthetic judgements claim universal validity, and the aesthetic basis of art, however, requires that the Kantian position is developed further. This is done by exploring his ideas concerning critical comparisons in the cultivation of taste, and art's relation to aesthetic ideas and genius. By linking earlier points to a more developed account of comparative critical factors, the Kantian approach offers a satisfying and comprehensive explanation of aesthetic experience and fine art. It is shown to also encompass some kinds of avant-garde work that were previously thought to limit its relevance.Less
This book explains the perceptual knowledge involved in aesthetic judgements. It does so by linking Kant's aesthetics to a critically upgraded account of his theory of knowledge. This upgraded theory emphasizes conceptual and imaginative structures, which Kant terms respectively, ‘categories’ and ‘schemata’. By describing examples of aesthetic judgement, it is shown that these judgements must involve categories and fundamental schemata (even though Kant himself, and most other commentators, have not fully appreciated the fact). It is argued, in turn, that this shows the aesthetic to be not just one kind of pleasurable experience amongst others, but one based on factors necessary to objective knowledge and personal identity, and one, indeed, which plays a role in how these capacities develop. The explanation of how individual aesthetic judgements claim universal validity, and the aesthetic basis of art, however, requires that the Kantian position is developed further. This is done by exploring his ideas concerning critical comparisons in the cultivation of taste, and art's relation to aesthetic ideas and genius. By linking earlier points to a more developed account of comparative critical factors, the Kantian approach offers a satisfying and comprehensive explanation of aesthetic experience and fine art. It is shown to also encompass some kinds of avant-garde work that were previously thought to limit its relevance.
Jonathan Wolff and Avner De-Shalit
- Published in print:
- 2007
- Published Online:
- May 2007
- ISBN:
- 9780199278268
- eISBN:
- 9780191707902
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199278268.003.0004
- Subject:
- Political Science, Political Theory
This chapter argues the importance of examining not only what functionings a person achieves, but also his or her prospect of securing and sustaining a level of functioning should he or she attempt ...
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This chapter argues the importance of examining not only what functionings a person achieves, but also his or her prospect of securing and sustaining a level of functioning should he or she attempt to do so. Key determinants of their prospects are: their probability of success, and what they have to sacrifice to achieve that probability of success. Thus, a key aspect of disadvantage is for one's functionings being or becoming insecure involuntarily, or when in order to secure certain functionings, one is forced to make other functionings insecure in a way that other people do not have to do. Accordingly, a policy of rectifying disadvantage would need to pay special attention to the question of how to secure functionings and how to avoid cross-category risk, i.e., when a risk spreads to other functionings. It is suggested that if a snapshot of a person's functioning is to be represented as a bar chart, the heights and widths of each bar should represent the functioning level achieved and the functioning's security, respectively.Less
This chapter argues the importance of examining not only what functionings a person achieves, but also his or her prospect of securing and sustaining a level of functioning should he or she attempt to do so. Key determinants of their prospects are: their probability of success, and what they have to sacrifice to achieve that probability of success. Thus, a key aspect of disadvantage is for one's functionings being or becoming insecure involuntarily, or when in order to secure certain functionings, one is forced to make other functionings insecure in a way that other people do not have to do. Accordingly, a policy of rectifying disadvantage would need to pay special attention to the question of how to secure functionings and how to avoid cross-category risk, i.e., when a risk spreads to other functionings. It is suggested that if a snapshot of a person's functioning is to be represented as a bar chart, the heights and widths of each bar should represent the functioning level achieved and the functioning's security, respectively.
Steve Awodey
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.003.0004
- Subject:
- Mathematics, Algebra
This chapter focuses on some of the connections between groups and categories. It focuses on three different aspects of the relationship between categories and groups: groups in a category, the ...
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This chapter focuses on some of the connections between groups and categories. It focuses on three different aspects of the relationship between categories and groups: groups in a category, the category of groups, and groups as categories. There are some exercises in the last part of the chapter.Less
This chapter focuses on some of the connections between groups and categories. It focuses on three different aspects of the relationship between categories and groups: groups in a category, the category of groups, and groups as categories. There are some exercises in the last part of the chapter.
Justin Willis
- Published in print:
- 1993
- Published Online:
- October 2011
- ISBN:
- 9780198203209
- eISBN:
- 9780191675782
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198203209.001.0001
- Subject:
- History, World Modern History
This is a history of the Kenyan city of Mombasa and its surrounding settlements from the mid-19th century to the height of colonial rule in the 1930s. The book places the island and town of Mombasa ...
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This is a history of the Kenyan city of Mombasa and its surrounding settlements from the mid-19th century to the height of colonial rule in the 1930s. The book places the island and town of Mombasa in its African context, incorporating the findings of historical and anthropological research. It examines the institutions and social networks that simultaneously united and divided the people of the region before the colonial period, demonstrating both their interdependence and the creation of distinct population categories. The book traces the development of these institutions under British rule, when the demands of the colonial economy caused officials to attempt far-reaching changes to the social structure and to physically remake the town of Mombasa. This is a re-interpretation of the history of Mombasa and its hinterland, based on archival research. It offers insights into the nature of ethnic identity.Less
This is a history of the Kenyan city of Mombasa and its surrounding settlements from the mid-19th century to the height of colonial rule in the 1930s. The book places the island and town of Mombasa in its African context, incorporating the findings of historical and anthropological research. It examines the institutions and social networks that simultaneously united and divided the people of the region before the colonial period, demonstrating both their interdependence and the creation of distinct population categories. The book traces the development of these institutions under British rule, when the demands of the colonial economy caused officials to attempt far-reaching changes to the social structure and to physically remake the town of Mombasa. This is a re-interpretation of the history of Mombasa and its hinterland, based on archival research. It offers insights into the nature of ethnic identity.
Ruth Garrett Millikan
- Published in print:
- 2005
- Published Online:
- February 2006
- ISBN:
- 9780199284764
- eISBN:
- 9780191603167
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199284768.001.0001
- Subject:
- Philosophy, Philosophy of Language
Guiding the work of most linguists and philosophers of language today is the assumption that language is governed by rules. This volume presents a different way of viewing the partial regularities ...
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Guiding the work of most linguists and philosophers of language today is the assumption that language is governed by rules. This volume presents a different way of viewing the partial regularities that language displays, the way they express norms and conventions. It argues that the central norms applying to language are non-evaluative; they are more like those norms of function and behavior that account for the survival and proliferation of biological species. Specific linguistic forms survive and are reproduced together with cooperative hearer responses because some portion of the time these patterns of production and response benefit both speakers and hearers. What needs to be reproduced, however, for a given language to survive is not specific conceptual rules or inference patterns, but only satisfaction conditions concerning distal objects and properties, and essential elements of hearer response. Thus, the psychological processes that support the use of proper names, of words for kinds, properties and so forth, need to be examined anew, resulting in a fairly uncompromising rejection of conceptual analysis as a tool in philosophy. Further results concern the distinction between the propositional content and the force of a linguistic utterance and a new description of illocutionary acts. It turns out that neither the intentionality of thought nor the intentionality of language is derived from the other. Also, the processes involved in understanding language are best modeled as a form of direct perception of the world parallel, for example, to perception mediated by the natural signs contained in structured light, and results in a radically new description of how children learn language.Less
Guiding the work of most linguists and philosophers of language today is the assumption that language is governed by rules. This volume presents a different way of viewing the partial regularities that language displays, the way they express norms and conventions. It argues that the central norms applying to language are non-evaluative; they are more like those norms of function and behavior that account for the survival and proliferation of biological species. Specific linguistic forms survive and are reproduced together with cooperative hearer responses because some portion of the time these patterns of production and response benefit both speakers and hearers. What needs to be reproduced, however, for a given language to survive is not specific conceptual rules or inference patterns, but only satisfaction conditions concerning distal objects and properties, and essential elements of hearer response. Thus, the psychological processes that support the use of proper names, of words for kinds, properties and so forth, need to be examined anew, resulting in a fairly uncompromising rejection of conceptual analysis as a tool in philosophy. Further results concern the distinction between the propositional content and the force of a linguistic utterance and a new description of illocutionary acts. It turns out that neither the intentionality of thought nor the intentionality of language is derived from the other. Also, the processes involved in understanding language are best modeled as a form of direct perception of the world parallel, for example, to perception mediated by the natural signs contained in structured light, and results in a radically new description of how children learn language.
Dov-Ber Kerler
- Published in print:
- 1993
- Published Online:
- October 2011
- ISBN:
- 9780198151661
- eISBN:
- 9780191672798
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198151661.003.0008
- Subject:
- Literature, European Literature
The eighteenth century represents a shift from the old to the new and modern literature. It involved the reprinting, adaption, and transition of the original forms, which occurred in various stages ...
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The eighteenth century represents a shift from the old to the new and modern literature. It involved the reprinting, adaption, and transition of the original forms, which occurred in various stages and in a systematic and organized fashion. Most examinations of this stage of reform have focused on the early and mid-eighteenth century, and little information is found towards the end of the century. To be able to understand it further, the chapter divides the changes that occurred into categories. Reprints are works that had only very minimal alteration, while re-editions have a larger number but more systemic change. Revision, on the other hand, is an extensive revision of the old form into a structure that fully adapts to modern concepts.Less
The eighteenth century represents a shift from the old to the new and modern literature. It involved the reprinting, adaption, and transition of the original forms, which occurred in various stages and in a systematic and organized fashion. Most examinations of this stage of reform have focused on the early and mid-eighteenth century, and little information is found towards the end of the century. To be able to understand it further, the chapter divides the changes that occurred into categories. Reprints are works that had only very minimal alteration, while re-editions have a larger number but more systemic change. Revision, on the other hand, is an extensive revision of the old form into a structure that fully adapts to modern concepts.
Masahiko Aoki
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199218530
- eISBN:
- 9780191711510
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199218530.003.0004
- Subject:
- Business and Management, Corporate Governance and Accountability, Strategy
Based on the recent development of epistemic game theory, this chapter attempts to resolve longstanding contested issues across social science disciplines about the nature and origin of institutions. ...
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Based on the recent development of epistemic game theory, this chapter attempts to resolve longstanding contested issues across social science disciplines about the nature and origin of institutions. These issues include: institutions as a pre-play design vs. spontaneous order, deontic constraints vs. rational choice, regularity of actions vs. shared meanings, endogenous and exogenous views, and so on. It argues that for a societal order to evolve through the recursive play of societal games, some social cognitive categories such as formal laws, norms, rules, and ritual, and organizations need to mediate between physical actions and the behavioral beliefs of individual players. Thus, pure methodological individualism must be laid to rest in institutional analysis. From such perspectives this chapter describes how institutions evolve and what the roles of business corporations can be in that process.Less
Based on the recent development of epistemic game theory, this chapter attempts to resolve longstanding contested issues across social science disciplines about the nature and origin of institutions. These issues include: institutions as a pre-play design vs. spontaneous order, deontic constraints vs. rational choice, regularity of actions vs. shared meanings, endogenous and exogenous views, and so on. It argues that for a societal order to evolve through the recursive play of societal games, some social cognitive categories such as formal laws, norms, rules, and ritual, and organizations need to mediate between physical actions and the behavioral beliefs of individual players. Thus, pure methodological individualism must be laid to rest in institutional analysis. From such perspectives this chapter describes how institutions evolve and what the roles of business corporations can be in that process.
E. J. Lowe
- Published in print:
- 2005
- Published Online:
- May 2006
- ISBN:
- 9780199254392
- eISBN:
- 9780191603600
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199254397.001.0001
- Subject:
- Philosophy, Metaphysics/Epistemology
The four-category ontology is a metaphysical system recognizing two fundamental categorial distinctions — these being between the particular and the universal, and between the substantial and the ...
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The four-category ontology is a metaphysical system recognizing two fundamental categorial distinctions — these being between the particular and the universal, and between the substantial and the non-substantial — which cut across each other to generate four fundamental ontological categories. The four categories thus generated are substantial particulars (‘objects’), non-substantial particulars (‘modes’), substantial universals (‘kinds’), and non-substantial universals (‘attributes’). This ontology has a lengthy pedigree, with many commentators attributing a version of it to Aristotle on the basis of certain passages in one of his early works, the Categories. Although it has been revived or rediscovered at various times during the history of western philosophy, it has never found widespread favour, perhaps due to its apparent lack of parsimony and its commitment to universals. In pursuit of ontological economy, metaphysicians have generally preferred to recognize fewer than four fundamental ontological categories. This book contends that the four-category ontology has an explanatory power which is unrivalled by more parsimonious systems, and that this counts decisively in its favour. It provides a uniquely powerful explanatory framework for a unified account of causation, dispositions, natural laws, natural necessity, and many other related matters, such as the semantics of counterfactual conditionals. The book is divided into four parts: the first setting out the framework of the four-category ontology, the second focusing on its central distinction between object and property, the third exploring its applications in the philosophy of natural science, and the fourth dealing with fundamental issues of truth and realism.Less
The four-category ontology is a metaphysical system recognizing two fundamental categorial distinctions — these being between the particular and the universal, and between the substantial and the non-substantial — which cut across each other to generate four fundamental ontological categories. The four categories thus generated are substantial particulars (‘objects’), non-substantial particulars (‘modes’), substantial universals (‘kinds’), and non-substantial universals (‘attributes’). This ontology has a lengthy pedigree, with many commentators attributing a version of it to Aristotle on the basis of certain passages in one of his early works, the Categories. Although it has been revived or rediscovered at various times during the history of western philosophy, it has never found widespread favour, perhaps due to its apparent lack of parsimony and its commitment to universals. In pursuit of ontological economy, metaphysicians have generally preferred to recognize fewer than four fundamental ontological categories. This book contends that the four-category ontology has an explanatory power which is unrivalled by more parsimonious systems, and that this counts decisively in its favour. It provides a uniquely powerful explanatory framework for a unified account of causation, dispositions, natural laws, natural necessity, and many other related matters, such as the semantics of counterfactual conditionals. The book is divided into four parts: the first setting out the framework of the four-category ontology, the second focusing on its central distinction between object and property, the third exploring its applications in the philosophy of natural science, and the fourth dealing with fundamental issues of truth and realism.
Steve Awodey
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.003.0008
- Subject:
- Mathematics, Algebra
This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it ...
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This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it is a straightforward generalization of facts that are fairly easily shown in relation to monoids and posets. The topics discussed include set-valued functor categories, Yoneda embedding, limits in categories of diagrams, colimits in categories of diagrams, exponentials in categories of diagrams, and Topoi. Exercises are provided in the last part of the chapter.Less
This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it is a straightforward generalization of facts that are fairly easily shown in relation to monoids and posets. The topics discussed include set-valued functor categories, Yoneda embedding, limits in categories of diagrams, colimits in categories of diagrams, exponentials in categories of diagrams, and Topoi. Exercises are provided in the last part of the chapter.
Richard Breen
- Published in print:
- 2004
- Published Online:
- November 2004
- ISBN:
- 9780199258451
- eISBN:
- 9780191601491
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199258457.003.0001
- Subject:
- Political Science, European Union
Outlines the objectives and layout of the volume; discusses the main theories that have guided empirical research on intergenerational social mobility and the findings of this research; and explains ...
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Outlines the objectives and layout of the volume; discusses the main theories that have guided empirical research on intergenerational social mobility and the findings of this research; and explains the main concepts and instruments (such as the class schema) that are used in the book.Less
Outlines the objectives and layout of the volume; discusses the main theories that have guided empirical research on intergenerational social mobility and the findings of this research; and explains the main concepts and instruments (such as the class schema) that are used in the book.
Alex Simpson
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198566519
- eISBN:
- 9780191713927
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566519.003.0003
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter advocates a pragmatic approach to constructive set theory, using axioms based solely on set-theoretic principles that are directly relevant to (constructive) mathematical practice. The ...
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This chapter advocates a pragmatic approach to constructive set theory, using axioms based solely on set-theoretic principles that are directly relevant to (constructive) mathematical practice. The aim is to leave the notion of set as unconstrained as possible, while remaining consistent with the ways in which sets are actually used in mathematical practice. Following this approach, the chapter presents theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories considered all have sound and complete classes of category-theoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar category-theoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.Less
This chapter advocates a pragmatic approach to constructive set theory, using axioms based solely on set-theoretic principles that are directly relevant to (constructive) mathematical practice. The aim is to leave the notion of set as unconstrained as possible, while remaining consistent with the ways in which sets are actually used in mathematical practice. Following this approach, the chapter presents theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories considered all have sound and complete classes of category-theoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar category-theoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.
E. J. Lowe
- Published in print:
- 2005
- Published Online:
- May 2006
- ISBN:
- 9780199254392
- eISBN:
- 9780191603600
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199254397.003.0001
- Subject:
- Philosophy, Metaphysics/Epistemology
A realist approach to metaphysics and ontology is defended in the face of some antirealist tendencies in contemporary philosophical thought. The general notion of an ontological category is explained ...
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A realist approach to metaphysics and ontology is defended in the face of some antirealist tendencies in contemporary philosophical thought. The general notion of an ontological category is explained and justified. Different systems of ontological categories are compared and contrasted with the four-category ontology: a one-category ontology of modes or tropes, a two-category ontology of particulars and universals, and a two-category ontology of substantial particulars and modes. The ontological status of states of affairs and natural laws, and the ontological implications of the truthmaker principle as advocated by D. M. Armstrong are discussed.Less
A realist approach to metaphysics and ontology is defended in the face of some antirealist tendencies in contemporary philosophical thought. The general notion of an ontological category is explained and justified. Different systems of ontological categories are compared and contrasted with the four-category ontology: a one-category ontology of modes or tropes, a two-category ontology of particulars and universals, and a two-category ontology of substantial particulars and modes. The ontological status of states of affairs and natural laws, and the ontological implications of the truthmaker principle as advocated by D. M. Armstrong are discussed.
E. J. Lowe
- Published in print:
- 2005
- Published Online:
- May 2006
- ISBN:
- 9780199254392
- eISBN:
- 9780191603600
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199254397.003.0002
- Subject:
- Philosophy, Metaphysics/Epistemology
The central principles of the four-category ontology are explained, especially its distinction between properties conceived as particulars (modes) and properties conceived as universals (attributes), ...
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The central principles of the four-category ontology are explained, especially its distinction between properties conceived as particulars (modes) and properties conceived as universals (attributes), and its distinction between substantial universals (kinds) and substantial particulars (objects). Its appeal to universals is defended and its account of the dispositional/occurrent distinction is explained. Some advantages of the four-category ontology over various of its more parsimonious rivals are sketched: its account of the individuation of tropes or modes, its analysis of laws, its analysis of dispositionality, and its account of property-perception.Less
The central principles of the four-category ontology are explained, especially its distinction between properties conceived as particulars (modes) and properties conceived as universals (attributes), and its distinction between substantial universals (kinds) and substantial particulars (objects). Its appeal to universals is defended and its account of the dispositional/occurrent distinction is explained. Some advantages of the four-category ontology over various of its more parsimonious rivals are sketched: its account of the individuation of tropes or modes, its analysis of laws, its analysis of dispositionality, and its account of property-perception.
E. J. Lowe
- Published in print:
- 2005
- Published Online:
- May 2006
- ISBN:
- 9780199254392
- eISBN:
- 9780191603600
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199254397.003.0003
- Subject:
- Philosophy, Metaphysics/Epistemology
The notion of a formal ontological relation is introduced and illustrated. Distinctions are drawn between various types of ontological dependence relations. The hierarchical character of systems of ...
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The notion of a formal ontological relation is introduced and illustrated. Distinctions are drawn between various types of ontological dependence relations. The hierarchical character of systems of ontological categories is examined, together with the ontological status of such categories themselves. It is argued that neither ontological categories nor formal ontological relations, such as instantiation and characterization, should be regarded as elements of being, that is, as entities in their own right. A distinction is drawn between form and content in ontology, paralleling but distinct from a similar distinction commonly made in logic.Less
The notion of a formal ontological relation is introduced and illustrated. Distinctions are drawn between various types of ontological dependence relations. The hierarchical character of systems of ontological categories is examined, together with the ontological status of such categories themselves. It is argued that neither ontological categories nor formal ontological relations, such as instantiation and characterization, should be regarded as elements of being, that is, as entities in their own right. A distinction is drawn between form and content in ontology, paralleling but distinct from a similar distinction commonly made in logic.
E. J. Lowe
- Published in print:
- 2005
- Published Online:
- May 2006
- ISBN:
- 9780199254392
- eISBN:
- 9780191603600
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199254397.003.0004
- Subject:
- Philosophy, Metaphysics/Epistemology
Traditional formal logic as developed by Fred Sommers is compared and contrasted with the modern quantified predicate logic that we owe to Gottlob Frege and Bertrand Russell; the latter is argued to ...
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Traditional formal logic as developed by Fred Sommers is compared and contrasted with the modern quantified predicate logic that we owe to Gottlob Frege and Bertrand Russell; the latter is argued to be implicitly committed to a two-category ontology of particulars and universals. A system of sortal logic is described, which exhibits some features of traditional formal logic and some of modern quantified predicate logic, such as its deployment of a symbol for identity. It is argued that this system represents more perspicuously than other systems the ontological distinctions of the four-category ontology, and that this counts as a distinct advantage in its favour.Less
Traditional formal logic as developed by Fred Sommers is compared and contrasted with the modern quantified predicate logic that we owe to Gottlob Frege and Bertrand Russell; the latter is argued to be implicitly committed to a two-category ontology of particulars and universals. A system of sortal logic is described, which exhibits some features of traditional formal logic and some of modern quantified predicate logic, such as its deployment of a symbol for identity. It is argued that this system represents more perspicuously than other systems the ontological distinctions of the four-category ontology, and that this counts as a distinct advantage in its favour.
